$If A = [\begin{array}{ccc} \cos α & - \sin α & 0 \\ \sin α & \cos α & 0 \\ 0 & 0 & 1 \end{array}], then (Adj A)^{- 1} =$

Moderate

A

$[\begin{matrix} \cos α & - \sin α & 0 \\ \sin α & \cos α & 0 \\ 0 & 0 & 1 \end{matrix}]$
B

$[\begin{matrix} - \cos α & - \sin α & 0 \\ \sin α & - \cos α & 0 \\ 0 & 0 & 1 \end{matrix}]$
C

$[\begin{matrix} - \cos α & \sin α & 0 \\ \sin α & - \cos α & 0 \\ 0 & 0 & 1 \end{matrix}]$
D

$[\begin{matrix} 1 & 0 & 0 \\ 0 & \cos α & - \sin α \\ 0 & \sin α & \cos α \end{matrix}]$

Solution

$\det A = \cos α (\cos α - 0) + \sin α (\sin α - 0) = \cos^{2} α + \sin^{2} α = 1$
$∴ A^{- 1} = A d j A \Rightarrow {(A d j A)}^{- 1} = {(A^{- 1})}^{- 1} = A$

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